Linked Questions

1
vote
0answers
56 views

How can I obtain a solution for the equation $a^2 + b^2 = c^2 + 1$? [duplicate]

For the equation $a^2 + b^2 = c^2$, the solution is: $a = m^2 - n^2, b= 2mn, c = m^2 + n^2$ $m,n\in\mathbb{Z}$ and $m > n$, free to choose How is a similar solution obtained for the equation $a^2 ...
17
votes
5answers
2k views

Integral solutions of $x^2+y^2+1=z^2$

I am interested in integral solutions of $$x^2+y^2+1=z^2.$$ Is there a complete theory comparable to the one for $x^2+y^2=z^2?$
5
votes
4answers
5k views

Solutions to $ax^2 + by^2 = cz^2$

The integer solutions to the equation $x^2 + y^2 = z^2$ are very well studied. I'm wondering if there's any literature about the integer solutions to the equation $ax^2 + by^2 = cz^2$ where a,b,c are ...
9
votes
4answers
1k views

Showing that $m^2-n^2+1$ is a square

Prove that if $m,n$ are odd integers such that $m^2-n^2+1$ divides $n^2-1$ then $m^2-n^2+1$ is a square number. I know that a solution can be obtained from Vieta jumping, but it seems very different ...
1
vote
2answers
6k views

Solution of Diophantine equation

Find all integral solutions of $x^2+1= y^2+z^2$. Actually I have to find all integral solution of $a(a+1)=b(b+1)+c(c+1)$. I reduced this in the above form I.e., $ (2a+1)^2+1= (2b+1)^2+(2c+1)^2$ .
4
votes
3answers
141 views

$x^2+y^2=z(4z+1)$ solutions

For a small project I am working on, I wish to find the solutions for $$x^2+y^2=z(4z+1)$$ in natural numbers $x,y,z$. I wish to automate finding solutions for $z$ up to a maximum value as efficient as ...
1
vote
3answers
165 views

$p^2+1=q^2+r^2$. Strange phenomenon of primes

Problem: Find prime solutions to the equation $p^2+1=q^2+r^2$ I welcome you to post your own solutions as well I have found a strange solution which I can't understand why it works(or what's the ...
0
votes
3answers
152 views

Solving $x^2 + y^2 = 1 + z^4$ with (x,y,z) = 1 and z < x < y

I have a computer programming problem where I need to find n many sets of integers that meet the condition $x^2 + y^2 = 1 + z^4$ with (x,y,z) = 1 and z < x < y I can do this relatively easily ...
0
votes
2answers
246 views

The diophantine equation $z^2=a^2+bx^2+cy^2$

Is there a way to obtain (enumerate) the integer solutions $(x,y,z)$ of the following quadratic Diophantine equation $z^2=a^2+bx^2+cy^2$ where $a$ is an integer and $b, c$ are positive integers? I ...
0
votes
0answers
187 views

Positive even numbered integer solutions of $y=n^2-m^2-x^2$

Prove that no integer $x$ exists where $y=n^2-m^2-x^2$ has solutions: For all even integer values of $y$ in the range $2\le y \le 2x+1$ where $x$ is odd. For all odd integer values of $y$ in the ...
0
votes
0answers
83 views

Number of integer points on a rotational hyperboloid of two sheets.

There are many integer points on the hyperboloid of two sheets $x^2+y^2-z^2=-1$. (0,0,1), (2,2,3), (4,8,9),... Let us denote such set as H. I will consider only the upper sheet $z>0$, but ...